The exterior Dirichlet problem for the homogeneous $k$-Hessian equation

We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is $|x|^{\frac{2k-n}{n}}+O(1)$ if $k>\frac{n}{2}$. By constructing smooth solutions of approximating non-degenerate $k$-Hessian equations with uniform $C^{1,1}$-estimates, we prove the existence part. The uniqueness follows from the comparison theorem and thus the $C^{1,1}$ regularity of the solution of the homogeneous $k$-Hessian equation in the exterior domain is proved. We also prove a uniform positive lower bound of the gradient. As an implication of the $C^{1,1}$ estimates, we derive an almost monotonicity formula along the level set of the approximating solution. In particular, we get an weighted geometric inequality which is a natural generalization of the $k=1$ case.